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Let :, (,) be a (left) group action of a Lie group on a smooth manifold ; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G {\displaystyle G} on M {\displaystyle M} consists of a Lie group homomorphism G → D i f f ( M ) {\displaystyle G\to \mathrm {Diff} (M)} .
A Morse–Bott function is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.)
Download as PDF; Printable version; In other projects Wikidata item; ... Pages in category "Smooth manifolds" The following 19 pages are in this category, out of 19 ...
In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to . More precisely, given p ∈ M {\displaystyle p\in M} one is interested in curves γ p : R → M {\displaystyle \gamma _{p}:\mathbb {R} \to M} such that:
Let be a smooth manifold and let be a one-parameter family of Riemannian or pseudo-Riemannian metrics. Suppose that it is a differentiable family in the sense that for any smooth coordinate chart, the derivatives v i j = ∂ ∂ t ( ( g t ) i j ) {\displaystyle v_{ij}={\frac {\partial }{\partial t}}{\big (}(g_{t})_{ij}{\big )}} exist and are ...
Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in R k to M. The group C k (M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain complex.
Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771. Introduction to Smooth Manifolds, Springer-Verlag, Graduate Texts in Mathematics, 2002, 2nd edition 2012 [6] Fredholm Operators and Einstein Metrics on Conformally Compact Manifolds.
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